Polytopal Bier spheres and nonrealizable central symmetries

Abstract

Bier spheres arise as deleted joins of simplicial complexes with their combinatorial Alexander duals and form one of the largest known families of simplicial spheres. We study centrally symmetric Bier spheres and give a simple criterion for when they cannot arise as boundaries of centrally symmetric polytopes. From this, we obtain a large new family of simplicial polytopes with combinatorial automorphisms that cannot be realized geometrically. Prior to our construction, the Bokowski--Ewald--Kleinschmidt polytope was the only known simplicial example exhibiting these properties. By Smith theory, these polytopes have noncontractible realization spaces. Finally, we establish that every Bier sphere with at most $12$ vertices is polytopal.